Homology Theory

Prove the $3 \times 3$ Lemma.

Consider the following commutative diagram of abelian groups

If all 3 columns and the first two rows are short exact, then the last row is also short exact.

Prove that if $M$ is a compact manifold of odd dimension, then $\chi (M) = 0$. Show examples of compact 4-manifolds with $\chi = 0, 1, 2, 3, 4$.

Let $X$ be a topological space. The suspension $\Sigma X$ of $X$ is defined to be the identification space obtained from $X \times [-1,1]$ by identifying $X \times \left\{ -1\right\}$ to a point and $X \times \left\{ 1\right\}$ to another point. For example the sphere $S^{n}$ is the suspension of $S^{n-1}$ with the north and south poles corresponding to the two identification points.

Compute the homology of $\Sigma X$ in terms of the homology of $X$.

Consider the chain complex

Compute $H_{n}(C)$ for all $n$.

Construct a CW complex which has the following $\mathbb {Z}$-homology groups:

Compute $H_{p}(S^{n} \vee S^{n} \vee \cdots \vee S^{n})$, $n \geq 0$, for all $p$.

Build a CW complex $X$ by adding two 2-cells to $S^{1}$, one by the map $z \to z^{4}$ and the other by the map $z \to z^{6}$. What is the homology of this space?

Let

Let the real projective plane $\mathbb {R}^{P^{2}}$ be embedded in the standard way in the real projective 5-space $\mathbb {R}^{P^{5}}$. Compute $H_{q}(\mathbb {R}^{P^{5}}/\mathbb {R}^{P^{2}})$, where $\mathbb {R}^{P^{5}}/\mathbb {R}^{P^{2}}$ is the space obtained from $\mathbb {R}^{P^{5}}$ by identifying $\mathbb {R}^{P^{2}}$ to a point.

Let

Prove that $Y \times \mathbb {R}$ is not homeomorphic to $\mathbb {R}^{2}$.

Let $A$ be a non-empty subset of $X$ and $X \cup CA$ be the union of $X$ with the cone of $A$; that is, $X \cup CA$ is obtained from the subset $X \times \left\{ 0\right\} \cup A \times [0,1]$ of $X \times [0,1]$ by identifying $A \times \left\{ 1\right\}$ to a point.

Prove that the reduced homology group $\tilde{H}_{n}(X \cup CA)$ is isomorphic to $H_{n}(X, A)$, for every $n$.

For any topological space $X$ let $\Sigma X$ denote its (unreduced) suspension. ($\Sigma X$ is the quotient space $(X \times [0,1])/{\sim }$, where $\sim$ denotes the equivalence relation generated by requiring that $(x,t) \sim (y,s)$ if $s = t = 0$ or $s = t = 1$.) If $f : X \to Y$ is a map, let $\Sigma f : \Sigma X \to \Sigma Y$ be the map of suspensions induced by the map $(x,t) \to (f(x), t)$ of $X \times [0,1]$.

Consider the following commutative diagram of abelian groups in which the row and column are exact sequences.

Suppose that $\gamma$ and $\beta_{0}$ are surjective. Prove that $\alpha_{3}$ is injective.

Suppose that a topological space $X$ is expressed as the union $U \cup V$ of two open path-connected subspaces such that $U \cap V$ is path connected, $H_{1}(U) = 0$, and the inclusion $U \cap V \to V$ induces a surjective homomorphism $H_{1}(U \cap V) \to H_{1}(V)$. (Here $H$ denotes singular homology.)

Let

denote the standard unit ball in Euclidean space $\mathbb {R}^{n}$, and

denote the standard $(n-1)$-sphere. Suppose that $f : D^{n} \to D^{n}$ is a continuous map such that the restriction of $f$ to $S^{n-1}$ is a homeomorphism from $S^{n-1}$ to $S^{n-1}$. Prove that $f$ is surjective.

Let $X$ be a connected CW complex with two 0-cells, three 1-cells, three 2-cells, and no higher-dimensional cells. Assume $H_{1}(X) \approx \mathbb {Z} \oplus \mathbb {Z}/3$. Compute the Euler characteristic of $X$ and determine (with proof) all possibilities for $H_{2}(X)$.

Let $X$ be a CW complex with exactly one $k{+}1$-cell. Prove that if $H_{k}(X; \mathbb {Z})$ is a non-trivial finite group then $H_{k+1}(X; \mathbb {Z}) = 0$.

Let

and

Let $X = A \cup B$. Compute the relative homology groups $H_{i}(X, X - (0,0,0,0))$ for all $i$.

Let $X$ be a path connected and locally path connected space and let $x \in X$. Let $Y = S^{1} \times S^{1} \times S^{2}$. Show that if $\pi_{1}(X, x)$ is finite, then any continuous map $f : X \to Y$ induces the trivial map $f_{*} : H_{i}(X, \mathbb {Z}) \to H_{i}(Y, \mathbb {Z})$ for all $i$ different from 0 and 2.

Call a commutative diagram of abelian groups

``exact'' if the sequence of groups and homomorphisms

It is an interesting fact that if

and

are exact, then so is

Prove exactness of

Let $X$ be a non-empty compact Hausdorff space and $f : X \to X$ be a continuous map. Prove that there exists a non-empty closed subset $A$ of $X$ such that $f(A) = A$. Give an example to show that compactness is essential for this assertion.

Recall that

($\mathbb {R}^{P^{n}}$ is the $n$-dimensional real projective space.) Prove that there is no function $f : S^{3} \to \mathbb {R}^{P^{3}}$ inducing an isomorphism on the third homology.

Let

Let $(B^{n}, S^{n-1})$ be the standard ball and sphere pair in $\mathbb {R}^{n}$, $n > 1$. Suppose that $f : (B^{n}, S^{n-1}) \to (X,A)$ is a continuous map and $f|_{S^{n-1}} : S^{n-1} \to A$ is a homeomorphism. Show that if $H_{n}(X) = 0$ then $H_{n}(X,A) = \mathbb {Z}$.

Let $A$ be a subspace of $S^{2} \times S^{2}$ homeomorphic to the 2-sphere $S^{2}$. State what the homology groups $H_{q}(S^{2} \times S^{2})$ are (no proof is required). What can you say about the possibilities for the relative homology groups $H_{q}(S^{2} \times S^{2}, A)$?

Compute the homology groups of the space $X$ obtained as the union of the 2-sphere $S^{2}$ and the $x$-axis in $\mathbb {R}^{3}$.

Define the unreduced suspension'' $\Sigma X$ of a space $X$ to be the quotient space of $I \times X$ obtained by identifying $\left\{ 0\right\} \times X$ to one point and $\left\{ 1\right\} \times X$ to one point. (This is the union of two cones'' on $X$.) Show that there is a natural isomorphism $\sigma_{X} : \tilde{H}_{i}(X) \to \tilde{H}_{i+1}(\Sigma X)$, for all $i \geq 0$.

Denote by $X$ the union of the torus $S^{1} \times S^{1}$ with the disc $D^{2}$, where $D^{2}$ is attached to $T^{2}$ by identifying $\partial D^{2}$ with a meridian curve $S^{1} \times \left\{ x_{0}\right\}$ in the torus, where $x_{0} \in S^{1}$.

Given homomorphism $h : A \to B$ and $g : C \to B$, the pull back of $h$ via $g$ is the group

Recall that if $C$ is a homeomorphic copy of the circle in $S^{3}$, then $H_{i}(S^{3} - C)$ is infinite cyclic for $i$ equal to 0 or 1 and is zero otherwise. Assuming this fact compute

It is known that if $X \subset S^{3}$ is homeomorphic to $S^{1}$ then $H_{*}(S^{3} - X) \approx H_{*}(S^{1})$. Use this fact to compute the homology of $S^{3} - Y$ where $Y$ is a subspace of $S^{3}$ homeomorphic to the disjoint union of two copies of $S^{1}$.

Compute the homology of $S^{1} \times S^{1} - \mathrm{point}$.

Suppose the following diagram is commutative, the rows are exact, and $\gamma_{n}$ is an isomorphism for all $n$.

Construct an exact sequence

Write out the proof of exactness at $B_{n}'$.

Suppose that $X$ is a space and $f : X \to Y$, $g : X \to Z$ are two maps of $X$ into contractible spaces $Y$ and $Z$. Let $M$ be the mapping cylinder of $f$ and $g$, that is, $M$ is the identification space obtained from the disjoint union of $Y$, $X \times I$ and $Z$ by identifying each $(x,0)$ with $f(x)$, and each $(x,1)$ with $g(x)$. Prove that $\tilde{H}_{q}M \cong \tilde{H}_{q-1}X$.

Let $\left\{ X_{i}, i = 1,2,3,\ldots \right\}$ be an infinite sequence of topological spaces and $f_{i} : X_{i} \to X_{i+1}$ be maps. The mapping telescope of this sequence is defined to be the identification space $T = \left(\prod_{i=1}^{\infty } X_{i} \times [0,1]\right)/{\sim }$, where $(x,1) \sim (y,0)$ if $x \in X_{i}$, $y \in X_{i+1}$ and $y = f_{i}(x)$ describes the identification. Consider the case where all $X_{i} = S^{m}$ ($m > 0$) and $f_{i} : X_{i} \to X_{i+1}$ is a map of degree $d$.

Let $R$ be a commutative ring with 1. Define an $R$-homology sphere to be a Hausdorff topological space $X$ such that $H_{i}(X;R) \cong H_{i}(S^{n}; R)$ for some sphere $S^{n}$.